Graph states of prime-power dimension from generalized CNOT quantum circuit

TL;DR

This paper constructs and classifies multipartite graph states of prime-power dimension using finite fields and generalized CNOT circuits, revealing maximally entangled states and simplifying their classification.

Contribution

It introduces a standard form for graph states of prime-power dimension, facilitating their classification and analysis of entanglement properties.

Findings

Standard form simplifies classification of graph states

Existence of multipartite maximally entangled states

Maximally entangled states for specific dimensions

Abstract

We construct multipartite graph states whose dimension is the power of a prime number. This is realized by the finite field, as well as the generalized controlled-NOT quantum circuit acting on two qudits. We propose the standard form of graph states up to local unitary transformations and particle permutations. The form greatly simplifies the classification of graph states as we illustrate up to five qudits. We also show that some graph states are multipartite maximally entangled states in the sense that any bipartite of the system produces a bipartite maximally entangled state. We further prove that 4-partite maximally entangled states exist when the dimension is an odd number at least three or a multiple of four.